Unidad2 - Informe 2
Ana Luisa Sotelo Ariza
2024-03-17
Ejercicio 2
La simulación ayuda a entender y validad las propiedades de los
estimadores estadísticos como son. insesgadez, eficiencia y la
consistencia principalmente. El siguiente problema permite evidenciar
las principales características de un grupo de estimadores propuestos
para la estimación de un parámetro asociado a un modelo de probabilidad.
Sean X1, X2, X3 y X4, una muestra aleatoria de tamaño n=4 cuya población
la conforma una distribución exponencial con parámetro θ desconocido.
Determine las características de cada uno de los siguientes estimadores
propuestos: 
Solución:
El proceso se realiza a partir de los siguientes pasos:
- Se crea una semilla.
- Se crea la función Estimadores la cual recibe el valor de theta y se encarga de lo siguiente: (Se calcula n en este caso 4 variables, se realiza una muestra, se caulculan las funciones de los estimadores, y se retorna los valores de los estimadores)
- Se asigna un valor a m quien almacenara el numero de veces de repeticiones para la función (20,50,100,1000).
- Se asigna un valor a theta (2)
- se invoca la función de Estimadores en donde se repitirá m (20,50,100,1000) veces y enviara el valor de theta (2)
- Se evalúa la media para las 4 variables en el total de repeticiones. (Propósito calcular la insesgadez).
- Se evalúa la desviación para las 4 variables en el total de repeticiones. (Propósito calcular la Eficiencia).
- Se presentara por cada repetición sus conclusiones independientes.
20 repeticiones:
set.seed(1)
#Semilla: fijar un valor inicial para el generador de números aleatorios
Estimadores = function(theta){
n=4
muestra <- rexp(n, rate=1/theta)
#Calculo de la muestra n=20, 50, 100, 1000
muestra
#Calculo estimadores
Estimadores1 <- ((muestra[1]+muestra[2])/6) + ((muestra[3] + muestra[4])/3)
Estimadores2 <- ((muestra[1] + (2*muestra[2]) + (3*muestra[3]) + (4*muestra[4]))/5)
Estimadores3 <- (sum(muestra))/4
Estimadores4 <- (min(muestra) + max(muestra))/2
return(Estimadores=cbind(Estimadores1,Estimadores2,Estimadores3,Estimadores4))
}
m <-20
theta = 2
resultado <- do.call(rbind,replicate(m,Estimadores(theta=theta),simplify = FALSE))
resultado## Estimadores1 Estimadores2 Estimadores3 Estimadores4
## [1,] 0.8359429 1.645907 1.111163 1.3214380
## [2,] 2.2898423 4.829369 2.550141 3.3310372
## [3,] 1.8030478 3.388394 1.628189 1.5377811
## [4,] 3.2803707 6.956031 3.875662 5.4591782
## [5,] 1.4605361 2.620099 1.728098 2.2129686
## [6,] 1.5754599 2.717649 1.933197 2.6586356
## [7,] 3.0802675 7.118728 2.351579 4.0183720
## [8,] 1.7050776 3.048747 1.821339 1.4725539
## [9,] 1.3656502 3.066552 1.435357 1.5239783
## [10,] 1.0000318 1.978763 1.006763 0.9865701
## [11,] 2.3996207 4.810233 2.326748 2.3205086
## [12,] 1.8104284 3.605922 1.571802 1.5944077
## [13,] 2.5746789 5.804663 2.561511 2.6010150
## [14,] 2.3065484 4.010754 2.673816 3.7756184
## [15,] 1.4265208 2.897329 1.199719 1.3158028
## [16,] 1.6121176 3.734307 1.508551 1.6616609
## [17,] 4.3327954 7.373428 4.565583 5.2639449
## [18,] 3.2816163 6.836626 2.873806 3.1258399
## [19,] 1.3590542 1.942261 1.818160 2.9619427
## [20,] 3.0215570 6.785174 2.745446 3.1111143cat("Media \n")## Mediaprint(prom<-apply(resultado,2, mean)) ## Estimadores1 Estimadores2 Estimadores3 Estimadores4
## 2.126058 4.258547 2.164331 2.612718# valores de las medias
cat("Desviaciones \n")## Desviacionesprint(desv<-apply(resultado,2, sd)) ## Estimadores1 Estimadores2 Estimadores3 Estimadores4
## 0.9051016 1.9182585 0.9068198 1.2862151# valores de la desviaciones estándar
boxplot(resultado, col = c("purple","pink","green","yellow"), ylim = c(0,10),main = "Estimadores " , ylab="Estimacion")
abline(h=theta,col='red') # línea indicando 1/2
grid()
Conclusiones:
Después de realizar 20 repeticiones en la primera ejecución, se pueden extraer las siguientes conclusiones sobre los resultados obtenidos de los 4 estimadores:
Respecto a la insesgadez, se observa que los estimadores 1 y 3 son los más cercanos al valor de theta (2). Sus resultados fueron 2.12 y 2.16 respectivamente, lo que sugiere una mayor aproximación al valor real.
En cuanto a la eficiencia, se encontró que los estimadores 1 y 3 presentan una menor varianza, con valores de 0.906 y 0.905 respectivamente. Esto indica una mayor consistencia en sus resultados y una menor dispersión con respecto al valor estimado.
El estimador 2 se identificó como el menos confiable debido a que tanto su insesgadez como su varianza muestran valores negativos. Su media fue de 4.25, alejándose considerablemente de theta, y una desviación de 1.91, la más alta entre los estimadores, lo que sugiere una mayor variabilidad en sus resultados.
En relación con el estimador 4, aunque sus valores no son alarmantes, no alcanzan la óptima precisión de los estimadores 1 y 3. Su media fue de 2.61 y la desviación de 1.28, lo que indica una menor consistencia y precisión en sus estimaciones.
50 repeticiones
set.seed(1)
#Semilla: fijar un valor inicial para el generador de números aleatorios
Estimadores = function(theta){
n=4
muestra <- rexp(n, rate=1/theta)
#Calculo de la muestra n=20, 50, 100, 1000
muestra
#Calculo estimadores
Estimadores1 <- ((muestra[1]+muestra[2])/6) + ((muestra[3] + muestra[4])/3)
Estimadores2 <- ((muestra[1] + (2*muestra[2]) + (3*muestra[3]) + (4*muestra[4]))/5)
Estimadores3 <- (sum(muestra))/4
Estimadores4 <- (min(muestra) + max(muestra))/2
return(Estimadores=cbind(Estimadores1,Estimadores2,Estimadores3,Estimadores4))
}
m <-50
theta = 2
resultado <- do.call(rbind,replicate(m,Estimadores(theta=theta),simplify = FALSE))
resultado## Estimadores1 Estimadores2 Estimadores3 Estimadores4
## [1,] 0.8359429 1.6459074 1.1111633 1.3214380
## [2,] 2.2898423 4.8293693 2.5501410 3.3310372
## [3,] 1.8030478 3.3883937 1.6281892 1.5377811
## [4,] 3.2803707 6.9560309 3.8756624 5.4591782
## [5,] 1.4605361 2.6200991 1.7280975 2.2129686
## [6,] 1.5754599 2.7176495 1.9331969 2.6586356
## [7,] 3.0802675 7.1187279 2.3515785 4.0183720
## [8,] 1.7050776 3.0487473 1.8213395 1.4725539
## [9,] 1.3656502 3.0665522 1.4353572 1.5239783
## [10,] 1.0000318 1.9787630 1.0067627 0.9865701
## [11,] 2.3996207 4.8102325 2.3267475 2.3205086
## [12,] 1.8104284 3.6059218 1.5718024 1.5944077
## [13,] 2.5746789 5.8046627 2.5615109 2.6010150
## [14,] 2.3065484 4.0107536 2.6738159 3.7756184
## [15,] 1.4265208 2.8973289 1.1997192 1.3158028
## [16,] 1.6121176 3.7343070 1.5085509 1.6616609
## [17,] 4.3327954 7.3734282 4.5655828 5.2639449
## [18,] 3.2816163 6.8366256 2.8738059 3.1258399
## [19,] 1.3590542 1.9422605 1.8181602 2.9619427
## [20,] 3.0215570 6.7851743 2.7454457 3.1111143
## [21,] 1.0500003 2.0381175 1.1361127 1.0675177
## [22,] 2.9076665 5.4313798 2.9527560 3.0880244
## [23,] 2.1370398 4.1765990 2.4642851 2.5478291
## [24,] 1.0647905 1.8655622 1.1247021 1.3044368
## [25,] 0.5400440 1.0644348 0.5693361 0.5791771
## [26,] 1.3577481 2.3190591 1.6223235 2.2794430
## [27,] 2.9781657 6.5695619 3.2191691 2.8897334
## [28,] 2.7151994 5.6611107 2.3686202 2.4013437
## [29,] 1.4796839 2.9685608 1.1504425 1.4567015
## [30,] 0.5103282 0.9649627 0.5090669 0.5052829
## [31,] 2.1207299 4.3115192 1.8352222 1.9455815
## [32,] 3.1488795 6.2454758 2.4113646 3.2795875
## [33,] 1.4382701 3.3102685 1.3992534 1.3445718
## [34,] 0.9296000 2.0354172 0.8867206 1.0153587
## [35,] 1.9910653 3.8863935 1.5571874 2.1725460
## [36,] 2.8096700 5.6314828 2.4966216 2.7793127
## [37,] 0.9949504 2.0699175 1.0024226 1.0248392
## [38,] 1.7725801 3.3678215 2.2837545 2.3515822
## [39,] 3.2942221 6.4393814 3.6184342 3.5699893
## [40,] 0.9080951 1.4231200 1.3182105 1.8693917
## [41,] 2.8708567 5.6419489 2.7717515 2.6040629
## [42,] 3.7550519 7.3547115 3.2423024 2.8928158
## [43,] 2.8150823 6.2359366 2.8474596 2.9445917
## [44,] 2.8074161 5.9690721 2.8059266 2.8014582
## [45,] 1.6730412 3.0662584 1.9086501 2.3726012
## [46,] 1.1816234 2.4038987 1.1362640 1.1925342
## [47,] 1.4786662 2.7165726 1.5735179 1.8402087
## [48,] 2.1389317 4.5928577 2.2335438 2.5173801
## [49,] 1.1993236 2.6885225 1.2376602 1.3526700
## [50,] 0.9505972 1.6615926 1.0287080 0.8692338cat("Media \n")## Mediaprint(prom<-apply(resultado,2, mean)) ## Estimadores1 Estimadores2 Estimadores3 Estimadores4
## 1.990810 3.985649 1.999968 2.262283# valores de las medias
cat("Desviaciones \n")## Desviacionesprint(desv<-apply(resultado,2, sd)) ## Estimadores1 Estimadores2 Estimadores3 Estimadores4
## 0.9074065 1.9011506 0.8803700 1.0750312# valores de la desviaciones estándar
boxplot(resultado, col = c("purple","pink","green","yellow"), ylim = c(0,10),main = "Estimadores " , ylab="Estimacion")
abline(h=theta,col='red') # línea indicando 1/2
grid()
Conclusiones:
Después de realizar 50 repeticiones en la primera ejecución, se pueden extraer las siguientes conclusiones sobre los resultados obtenidos de los 4 estimadores:
Respecto a la insesgadez, se observa que los estimadores 1 y 3 son los más cercanos al valor de theta (2). Sus resultados fueron 1.990 y 1.999 respectivamente, lo que sugiere una mayor aproximación al valor real.
En cuanto a la eficiencia, se encontró que los estimadores 1 y 3 presentan una menor varianza, con valores de 0.90 y 0.88 respectivamente. Esto indica una mayor consistencia en sus resultados y una menor dispersión con respecto al valor estimado.
Entre el estimador 1 y 3, el estimador 3 cuenta con mejores resultados tanto en su valor mas cercano a theta como en varianza entre los valores.
El estimador 2 se identificó como el menos confiable debido a que tanto su insesgadez como su varianza muestran valores negativos. Su media fue de 3.98, alejándose considerablemente de theta, y una desviación de 1.90, la más alta entre los estimadores, lo que sugiere una mayor variabilidad en sus resultados.
En relación con el estimador 4, aunque sus valores no son alarmantes, no alcanzan la óptima precisión de los estimadores 1 y 3. Su media fue de 2.26 y la desviación de 1.07, lo que indica una menor consistencia y precisión en sus estimaciones.
100 repeticiones
set.seed(1)
#Semilla: fijar un valor inicial para el generador de números aleatorios
Estimadores = function(theta){
n=4
muestra <- rexp(n, rate=1/theta)
#Calculo de la muestra n=20, 50, 100, 1000
muestra
#Calculo estimadores
Estimadores1 <- ((muestra[1]+muestra[2])/6) + ((muestra[3] + muestra[4])/3)
Estimadores2 <- ((muestra[1] + (2*muestra[2]) + (3*muestra[3]) + (4*muestra[4]))/5)
Estimadores3 <- (sum(muestra))/4
Estimadores4 <- (min(muestra) + max(muestra))/2
return(Estimadores=cbind(Estimadores1,Estimadores2,Estimadores3,Estimadores4))
}
m <-100
theta = 2
resultado <- do.call(rbind,replicate(m,Estimadores(theta=theta),simplify = FALSE))
resultado## Estimadores1 Estimadores2 Estimadores3 Estimadores4
## [1,] 0.8359429 1.6459074 1.1111633 1.3214380
## [2,] 2.2898423 4.8293693 2.5501410 3.3310372
## [3,] 1.8030478 3.3883937 1.6281892 1.5377811
## [4,] 3.2803707 6.9560309 3.8756624 5.4591782
## [5,] 1.4605361 2.6200991 1.7280975 2.2129686
## [6,] 1.5754599 2.7176495 1.9331969 2.6586356
## [7,] 3.0802675 7.1187279 2.3515785 4.0183720
## [8,] 1.7050776 3.0487473 1.8213395 1.4725539
## [9,] 1.3656502 3.0665522 1.4353572 1.5239783
## [10,] 1.0000318 1.9787630 1.0067627 0.9865701
## [11,] 2.3996207 4.8102325 2.3267475 2.3205086
## [12,] 1.8104284 3.6059218 1.5718024 1.5944077
## [13,] 2.5746789 5.8046627 2.5615109 2.6010150
## [14,] 2.3065484 4.0107536 2.6738159 3.7756184
## [15,] 1.4265208 2.8973289 1.1997192 1.3158028
## [16,] 1.6121176 3.7343070 1.5085509 1.6616609
## [17,] 4.3327954 7.3734282 4.5655828 5.2639449
## [18,] 3.2816163 6.8366256 2.8738059 3.1258399
## [19,] 1.3590542 1.9422605 1.8181602 2.9619427
## [20,] 3.0215570 6.7851743 2.7454457 3.1111143
## [21,] 1.0500003 2.0381175 1.1361127 1.0675177
## [22,] 2.9076665 5.4313798 2.9527560 3.0880244
## [23,] 2.1370398 4.1765990 2.4642851 2.5478291
## [24,] 1.0647905 1.8655622 1.1247021 1.3044368
## [25,] 0.5400440 1.0644348 0.5693361 0.5791771
## [26,] 1.3577481 2.3190591 1.6223235 2.2794430
## [27,] 2.9781657 6.5695619 3.2191691 2.8897334
## [28,] 2.7151994 5.6611107 2.3686202 2.4013437
## [29,] 1.4796839 2.9685608 1.1504425 1.4567015
## [30,] 0.5103282 0.9649627 0.5090669 0.5052829
## [31,] 2.1207299 4.3115192 1.8352222 1.9455815
## [32,] 3.1488795 6.2454758 2.4113646 3.2795875
## [33,] 1.4382701 3.3102685 1.3992534 1.3445718
## [34,] 0.9296000 2.0354172 0.8867206 1.0153587
## [35,] 1.9910653 3.8863935 1.5571874 2.1725460
## [36,] 2.8096700 5.6314828 2.4966216 2.7793127
## [37,] 0.9949504 2.0699175 1.0024226 1.0248392
## [38,] 1.7725801 3.3678215 2.2837545 2.3515822
## [39,] 3.2942221 6.4393814 3.6184342 3.5699893
## [40,] 0.9080951 1.4231200 1.3182105 1.8693917
## [41,] 2.8708567 5.6419489 2.7717515 2.6040629
## [42,] 3.7550519 7.3547115 3.2423024 2.8928158
## [43,] 2.8150823 6.2359366 2.8474596 2.9445917
## [44,] 2.8074161 5.9690721 2.8059266 2.8014582
## [45,] 1.6730412 3.0662584 1.9086501 2.3726012
## [46,] 1.1816234 2.4038987 1.1362640 1.1925342
## [47,] 1.4786662 2.7165726 1.5735179 1.8402087
## [48,] 2.1389317 4.5928577 2.2335438 2.5173801
## [49,] 1.1993236 2.6885225 1.2376602 1.3526700
## [50,] 0.9505972 1.6615926 1.0287080 0.8692338
## [51,] 1.3508805 2.9419307 1.2886309 1.1018821
## [52,] 2.0314872 3.9187788 1.7354991 2.3919762
## [53,] 1.4830582 3.3387801 1.4607071 1.5277603
## [54,] 1.2134754 2.0677427 1.1976196 1.2070870
## [55,] 2.0433968 4.0246909 2.2437729 2.2888925
## [56,] 1.8284508 3.5776451 1.7228375 1.4190829
## [57,] 2.2884019 4.1177989 2.1880052 2.1846093
## [58,] 3.9726892 9.1306874 3.3850061 5.0334017
## [59,] 0.9126923 1.8570405 0.8702705 0.8335528
## [60,] 1.5625786 3.1292849 1.4961083 1.4399572
## [61,] 3.3746540 6.9600383 3.1816715 3.5155144
## [62,] 2.1010989 4.7798960 1.7336615 2.8104696
## [63,] 3.1364127 5.9094975 2.6229743 3.9206406
## [64,] 0.9366993 1.8126334 0.9202283 0.8708154
## [65,] 1.0517786 1.9506347 1.4250697 1.4340846
## [66,] 2.1335322 4.4031721 2.8305449 3.2183599
## [67,] 1.1419010 2.0407698 1.0689029 1.1256576
## [68,] 2.5281273 4.9609039 2.7171256 2.7665040
## [69,] 5.6279991 12.5822398 4.3240900 6.3357852
## [70,] 2.8673935 4.7974517 3.1593143 4.0350768
## [71,] 1.7116025 3.7044034 2.2460035 3.4728301
## [72,] 0.7544788 1.4197239 0.9011341 0.9775632
## [73,] 1.0236902 2.1667310 1.3474887 1.9696639
## [74,] 1.1543176 2.0932292 1.1933001 1.3102477
## [75,] 2.0143218 3.9012484 2.1358078 1.9388059
## [76,] 3.4595546 7.2910307 3.6143026 4.0785467
## [77,] 1.6812801 2.8110576 1.7657254 2.0033826
## [78,] 1.8282095 3.5527252 2.4476082 2.4663123
## [79,] 4.0547133 8.6945818 3.1092304 3.3691530
## [80,] 1.6059128 3.4734746 1.5296779 1.7583826
## [81,] 0.8114863 1.4564323 0.9893972 1.2278367
## [82,] 2.5025614 5.3992135 2.1084060 2.6994775
## [83,] 1.3431845 2.8272705 1.1839000 1.3768692
## [84,] 1.1269286 1.9961387 1.1351441 1.1104975
## [85,] 2.0930333 4.0222035 1.9566040 2.3658919
## [86,] 1.5763130 3.5108543 1.7376911 2.1352873
## [87,] 1.4363477 2.7443598 1.7831823 1.8091778
## [88,] 0.4797127 1.0952929 0.6477142 1.1439168
## [89,] 2.3570335 4.8416663 2.2042114 2.0380354
## [90,] 2.0958437 4.8971314 2.3652060 2.9989073
## [91,] 1.0694720 1.8828591 1.1107941 1.2347603
## [92,] 1.5787339 3.1460443 1.6590951 1.6072928
## [93,] 1.1957343 1.8881818 1.5209920 2.3269680
## [94,] 2.0475241 3.5501493 2.1823721 2.3723735
## [95,] 1.4683870 2.8316981 1.5035878 1.3979855
## [96,] 1.9655750 3.6061765 2.6087919 2.7780085
## [97,] 1.3019001 2.9511341 1.7958008 3.1580365
## [98,] 0.7601330 1.6693939 0.7898762 0.7006466
## [99,] 1.6961545 3.4998777 1.2865414 1.3765501
## [100,] 1.9876238 3.2471725 2.1976133 2.3022714cat("Media \n")## Mediaprint(prom<-apply(resultado,2, mean)) ## Estimadores1 Estimadores2 Estimadores3 Estimadores4
## 1.933090 3.877555 1.946277 2.240810# valores de las medias
cat("Desviaciones \n")## Desviacionesprint(desv<-apply(resultado,2, sd)) ## Estimadores1 Estimadores2 Estimadores3 Estimadores4
## 0.9353153 2.0271779 0.8413133 1.1066431# valores de la desviaciones estándar
boxplot(resultado, col = c("purple","pink","green","yellow"), ylim = c(0,10),main = "Estimadores " , ylab="Estimacion")
abline(h=theta,col='red') # línea indicando 1/2
grid()
Conclusiones:
Después de realizar 100 repeticiones en la primera ejecución, se pueden extraer las siguientes conclusiones sobre los resultados obtenidos de los 4 estimadores:
Respecto a la insesgadez, se observa que los estimadores 1 y 3 son los más cercanos al valor de theta (2). Sus resultados fueron 1.93 y 1.94 respectivamente, lo que sugiere una mayor aproximación al valor real.
En cuanto a la eficiencia, se encontró que los estimadores 1 y 3 presentan una menor varianza, con valores de 0.93 y 0.84 respectivamente. Esto indica una mayor consistencia en sus resultados y una menor dispersión con respecto al valor estimado.
El estimador 2 se identificó como el menos confiable debido a que tanto su insesgadez como su varianza muestran valores negativos. Su media fue de 3.87, alejándose considerablemente de theta, y una desviación de 2.02, la más alta entre los estimadores, lo que sugiere una mayor variabilidad en sus resultados.
En relación con el estimador 4, aunque sus valores no son alarmantes, no alcanzan la óptima precisión de los estimadores 1 y 3. Su media fue de 2.24 y la desviación de 1.10, lo que indica una menor consistencia y precisión en sus estimaciones.
1000 repeticiones
set.seed(1)
#Semilla: fijar un valor inicial para el generador de números aleatorios
Estimadores = function(theta){
n=4
muestra <- rexp(n, rate=1/theta)
#Calculo de la muestra n=20, 50, 100, 1000
muestra
#Calculo estimadores
Estimadores1 <- ((muestra[1]+muestra[2])/6) + ((muestra[3] + muestra[4])/3)
Estimadores2 <- ((muestra[1] + (2*muestra[2]) + (3*muestra[3]) + (4*muestra[4]))/5)
Estimadores3 <- (sum(muestra))/4
Estimadores4 <- (min(muestra) + max(muestra))/2
return(Estimadores=cbind(Estimadores1,Estimadores2,Estimadores3,Estimadores4))
}
m <-1000
theta = 2
resultado <- do.call(rbind,replicate(m,Estimadores(theta=theta),simplify = FALSE))
resultado## Estimadores1 Estimadores2 Estimadores3 Estimadores4
## [1,] 0.8359429 1.6459074 1.1111633 1.3214380
## [2,] 2.2898423 4.8293693 2.5501410 3.3310372
## [3,] 1.8030478 3.3883937 1.6281892 1.5377811
## [4,] 3.2803707 6.9560309 3.8756624 5.4591782
## [5,] 1.4605361 2.6200991 1.7280975 2.2129686
## [6,] 1.5754599 2.7176495 1.9331969 2.6586356
## [7,] 3.0802675 7.1187279 2.3515785 4.0183720
## [8,] 1.7050776 3.0487473 1.8213395 1.4725539
## [9,] 1.3656502 3.0665522 1.4353572 1.5239783
## [10,] 1.0000318 1.9787630 1.0067627 0.9865701
## [11,] 2.3996207 4.8102325 2.3267475 2.3205086
## [12,] 1.8104284 3.6059218 1.5718024 1.5944077
## [13,] 2.5746789 5.8046627 2.5615109 2.6010150
## [14,] 2.3065484 4.0107536 2.6738159 3.7756184
## [15,] 1.4265208 2.8973289 1.1997192 1.3158028
## [16,] 1.6121176 3.7343070 1.5085509 1.6616609
## [17,] 4.3327954 7.3734282 4.5655828 5.2639449
## [18,] 3.2816163 6.8366256 2.8738059 3.1258399
## [19,] 1.3590542 1.9422605 1.8181602 2.9619427
## [20,] 3.0215570 6.7851743 2.7454457 3.1111143
## [21,] 1.0500003 2.0381175 1.1361127 1.0675177
## [22,] 2.9076665 5.4313798 2.9527560 3.0880244
## [23,] 2.1370398 4.1765990 2.4642851 2.5478291
## [24,] 1.0647905 1.8655622 1.1247021 1.3044368
## [25,] 0.5400440 1.0644348 0.5693361 0.5791771
## [26,] 1.3577481 2.3190591 1.6223235 2.2794430
## [27,] 2.9781657 6.5695619 3.2191691 2.8897334
## [28,] 2.7151994 5.6611107 2.3686202 2.4013437
## [29,] 1.4796839 2.9685608 1.1504425 1.4567015
## [30,] 0.5103282 0.9649627 0.5090669 0.5052829
## [31,] 2.1207299 4.3115192 1.8352222 1.9455815
## [32,] 3.1488795 6.2454758 2.4113646 3.2795875
## [33,] 1.4382701 3.3102685 1.3992534 1.3445718
## [34,] 0.9296000 2.0354172 0.8867206 1.0153587
## [35,] 1.9910653 3.8863935 1.5571874 2.1725460
## [36,] 2.8096700 5.6314828 2.4966216 2.7793127
## [37,] 0.9949504 2.0699175 1.0024226 1.0248392
## [38,] 1.7725801 3.3678215 2.2837545 2.3515822
## [39,] 3.2942221 6.4393814 3.6184342 3.5699893
## [40,] 0.9080951 1.4231200 1.3182105 1.8693917
## [41,] 2.8708567 5.6419489 2.7717515 2.6040629
## [42,] 3.7550519 7.3547115 3.2423024 2.8928158
## [43,] 2.8150823 6.2359366 2.8474596 2.9445917
## [44,] 2.8074161 5.9690721 2.8059266 2.8014582
## [45,] 1.6730412 3.0662584 1.9086501 2.3726012
## [46,] 1.1816234 2.4038987 1.1362640 1.1925342
## [47,] 1.4786662 2.7165726 1.5735179 1.8402087
## [48,] 2.1389317 4.5928577 2.2335438 2.5173801
## [49,] 1.1993236 2.6885225 1.2376602 1.3526700
## [50,] 0.9505972 1.6615926 1.0287080 0.8692338
## [51,] 1.3508805 2.9419307 1.2886309 1.1018821
## [52,] 2.0314872 3.9187788 1.7354991 2.3919762
## [53,] 1.4830582 3.3387801 1.4607071 1.5277603
## [54,] 1.2134754 2.0677427 1.1976196 1.2070870
## [55,] 2.0433968 4.0246909 2.2437729 2.2888925
## [56,] 1.8284508 3.5776451 1.7228375 1.4190829
## [57,] 2.2884019 4.1177989 2.1880052 2.1846093
## [58,] 3.9726892 9.1306874 3.3850061 5.0334017
## [59,] 0.9126923 1.8570405 0.8702705 0.8335528
## [60,] 1.5625786 3.1292849 1.4961083 1.4399572
## [61,] 3.3746540 6.9600383 3.1816715 3.5155144
## [62,] 2.1010989 4.7798960 1.7336615 2.8104696
## [63,] 3.1364127 5.9094975 2.6229743 3.9206406
## [64,] 0.9366993 1.8126334 0.9202283 0.8708154
## [65,] 1.0517786 1.9506347 1.4250697 1.4340846
## [66,] 2.1335322 4.4031721 2.8305449 3.2183599
## [67,] 1.1419010 2.0407698 1.0689029 1.1256576
## [68,] 2.5281273 4.9609039 2.7171256 2.7665040
## [69,] 5.6279991 12.5822398 4.3240900 6.3357852
## [70,] 2.8673935 4.7974517 3.1593143 4.0350768
## [71,] 1.7116025 3.7044034 2.2460035 3.4728301
## [72,] 0.7544788 1.4197239 0.9011341 0.9775632
## [73,] 1.0236902 2.1667310 1.3474887 1.9696639
## [74,] 1.1543176 2.0932292 1.1933001 1.3102477
## [75,] 2.0143218 3.9012484 2.1358078 1.9388059
## [76,] 3.4595546 7.2910307 3.6143026 4.0785467
## [77,] 1.6812801 2.8110576 1.7657254 2.0033826
## [78,] 1.8282095 3.5527252 2.4476082 2.4663123
## [79,] 4.0547133 8.6945818 3.1092304 3.3691530
## [80,] 1.6059128 3.4734746 1.5296779 1.7583826
## [81,] 0.8114863 1.4564323 0.9893972 1.2278367
## [82,] 2.5025614 5.3992135 2.1084060 2.6994775
## [83,] 1.3431845 2.8272705 1.1839000 1.3768692
## [84,] 1.1269286 1.9961387 1.1351441 1.1104975
## [85,] 2.0930333 4.0222035 1.9566040 2.3658919
## [86,] 1.5763130 3.5108543 1.7376911 2.1352873
## [87,] 1.4363477 2.7443598 1.7831823 1.8091778
## [88,] 0.4797127 1.0952929 0.6477142 1.1439168
## [89,] 2.3570335 4.8416663 2.2042114 2.0380354
## [90,] 2.0958437 4.8971314 2.3652060 2.9989073
## [91,] 1.0694720 1.8828591 1.1107941 1.2347603
## [92,] 1.5787339 3.1460443 1.6590951 1.6072928
## [93,] 1.1957343 1.8881818 1.5209920 2.3269680
## [94,] 2.0475241 3.5501493 2.1823721 2.3723735
## [95,] 1.4683870 2.8316981 1.5035878 1.3979855
## [96,] 1.9655750 3.6061765 2.6087919 2.7780085
## [97,] 1.3019001 2.9511341 1.7958008 3.1580365
## [98,] 0.7601330 1.6693939 0.7898762 0.7006466
## [99,] 1.6961545 3.4998777 1.2865414 1.3765501
## [100,] 1.9876238 3.2471725 2.1976133 2.3022714
## [101,] 1.3612851 2.6942735 1.0725056 1.2492318
## [102,] 1.3464978 2.0352667 1.6011796 2.3652251
## [103,] 1.4895971 3.0114176 1.8194098 2.0562238
## [104,] 2.4247068 4.7774848 1.9808212 2.5474183
## [105,] 1.2836286 2.6003208 1.1810790 1.2009696
## [106,] 4.7184487 10.4870516 4.2744158 5.6065146
## [107,] 0.7698215 1.6721999 0.6746582 0.9108872
## [108,] 2.7510605 5.2051026 2.2129916 3.4936068
## [109,] 2.6513415 5.7457480 2.4162514 3.1215218
## [110,] 1.6109715 2.5774161 2.2646603 3.2222886
## [111,] 2.2669395 4.6413303 2.4018990 2.8054866
## [112,] 2.0603315 4.4735739 1.8287180 2.0487368
## [113,] 1.9546267 4.4200501 2.0175557 1.8287689
## [114,] 0.8426415 1.8377435 0.7782061 0.7710979
## [115,] 2.7714458 4.4177566 2.9657339 3.5485979
## [116,] 1.7083268 3.0711491 1.7688629 1.9504712
## [117,] 2.3450438 4.3238446 2.5202982 3.0460615
## [118,] 1.2209514 2.2997127 1.1932328 1.1100772
## [119,] 1.2326979 2.8079974 1.2843108 1.1294723
## [120,] 1.0770169 1.9817892 1.3984070 1.4098912
## [121,] 0.9040454 1.9526095 1.0588123 1.1974980
## [122,] 1.6060675 3.1343872 1.3706147 1.1858998
## [123,] 1.5091508 3.2238390 1.3456427 1.6959784
## [124,] 1.1140307 2.4720671 1.4492236 2.3944304
## [125,] 1.3569705 2.7961344 1.2699815 1.3884761
## [126,] 1.2478906 2.8045239 0.9509908 1.4259478
## [127,] 3.3338649 7.0142267 3.3168018 3.2656125
## [128,] 3.8854507 8.9644973 3.4454102 4.7655318
## [129,] 1.7416356 3.4801160 1.7898040 1.8300847
## [130,] 5.5087333 12.3494563 4.4629238 6.7470881
## [131,] 5.4443527 10.2787667 5.2300009 5.8730565
## [132,] 3.7250333 7.4263326 4.4161193 5.0478704
## [133,] 2.6522612 5.3953123 2.3186844 2.4252863
## [134,] 3.3995029 6.2722155 3.5043244 3.1898598
## [135,] 4.1624499 8.9006312 3.3638298 4.0045384
## [136,] 0.5718338 0.9612641 0.7314434 0.8295494
## [137,] 1.1790226 2.6486995 1.2085962 1.1198753
## [138,] 1.5339688 2.9233296 1.6699366 1.8954887
## [139,] 1.6925254 2.8492586 1.7911843 2.0871609
## [140,] 0.9357076 2.1059613 1.1703947 1.8744561
## [141,] 2.7921202 6.1950966 3.2305906 4.5460018
## [142,] 1.6259887 3.3915625 1.8539290 2.0090310
## [143,] 2.4657632 4.9705609 3.3896128 4.2426776
## [144,] 1.8933558 3.6373112 1.9426219 1.7948235
## [145,] 0.7956595 1.8371165 0.8703064 1.0942471
## [146,] 2.1456401 4.2742283 1.8826281 2.1882697
## [147,] 0.9793684 1.8353247 0.8423611 1.2533828
## [148,] 0.7136569 1.4377294 0.5767365 0.6876179
## [149,] 1.5573371 3.1352521 1.2340954 1.5058924
## [150,] 3.6323276 5.8768385 4.2136169 5.9574849
## [151,] 1.2971866 2.4101006 1.5348976 1.7578679
## [152,] 1.2316210 2.4562782 1.2927433 1.2490303
## [153,] 1.5147324 3.0699368 1.6981462 1.9827399
## [154,] 0.7533191 1.6741599 1.0139953 1.6934759
## [155,] 1.8008437 3.4996163 2.1544458 2.0560216
## [156,] 2.2623028 4.6179671 1.8954230 1.7619468
## [157,] 2.2538628 5.1619560 2.1427665 2.4760554
## [158,] 1.5765765 2.6830886 2.2775737 2.7517378
## [159,] 2.9383878 6.8955660 3.6368037 5.7320513
## [160,] 2.8233284 6.2179558 3.1373402 4.0793758
## [161,] 1.7167954 3.8523501 2.0634363 3.1033591
## [162,] 1.9619368 4.3583902 2.0666506 2.3807920
## [163,] 3.4620510 6.3289295 2.8346152 4.7169228
## [164,] 2.1978160 4.1861686 2.5254156 2.3293569
## [165,] 2.2374919 5.0688530 2.8784324 4.7929583
## [166,] 1.2377682 2.7748790 1.1634359 0.9404392
## [167,] 1.7781868 3.7815468 2.0759388 2.9366239
## [168,] 1.0507155 2.0393890 1.0315891 1.0889682
## [169,] 1.3452346 2.5739441 1.5422464 1.6365476
## [170,] 1.9827949 3.8685189 2.2482695 2.2710668
## [171,] 1.4410932 2.7591477 1.6507812 1.7486898
## [172,] 2.7459352 5.1751693 2.4088643 3.3476285
## [173,] 3.7472594 7.9852910 3.5061384 4.2295012
## [174,] 1.9376282 3.7325673 2.4281224 2.5618941
## [175,] 1.1968271 2.1549949 1.2148311 1.2688433
## [176,] 1.7241501 3.3349851 1.8187752 2.0074382
## [177,] 2.1840239 4.1018029 2.2787845 2.5630662
## [178,] 2.9193941 5.8673657 2.6972444 2.7397142
## [179,] 3.0618275 6.4029428 3.2452975 3.7957076
## [180,] 1.6748352 3.7569073 1.5466985 1.5616113
## [181,] 1.1639526 2.4045836 1.1103193 1.2483821
## [182,] 2.3132126 5.0362340 2.3918362 2.6277070
## [183,] 2.5970675 4.8086573 1.9983996 3.5607621
## [184,] 3.1319054 6.6785060 2.7815084 3.0115393
## [185,] 4.8339414 9.7024088 3.9536389 3.8705169
## [186,] 1.6563844 3.7021747 1.6375840 1.5811828
## [187,] 2.0086951 4.4652815 2.2464376 2.9596654
## [188,] 2.7684085 5.5561411 2.3499927 2.6444620
## [189,] 2.2043309 3.8704136 2.7045837 2.3709609
## [190,] 1.0933280 2.2381198 0.9604112 0.9646687
## [191,] 1.9813072 3.4302108 1.9107642 1.6991353
## [192,] 1.0569376 1.6843697 1.3708156 1.7119094
## [193,] 1.3881639 2.6578716 1.4521390 1.6440642
## [194,] 1.2469219 2.4147963 1.2028345 1.0893317
## [195,] 2.4569063 5.3273417 2.3179221 2.7348747
## [196,] 0.6247919 1.3980819 0.4897398 0.7259333
## [197,] 0.7502022 1.6438286 0.6479359 0.5669482
## [198,] 1.9200295 3.8010456 2.2377063 2.2113456
## [199,] 2.1477067 4.0930778 2.0357906 2.3715388
## [200,] 3.7841876 7.1982618 3.8321878 3.6881872
## [201,] 0.8969326 1.4742440 0.9851853 1.2499435
## [202,] 2.1201428 4.1992817 1.6527892 2.2226529
## [203,] 4.0663418 8.3441962 4.0360978 3.9976125
## [204,] 3.1306511 6.3020755 3.1778940 3.0361651
## [205,] 3.2389396 6.4713287 3.5085091 3.2553161
## [206,] 2.1169127 4.2562109 1.8535842 1.6462194
## [207,] 2.3342216 4.6457091 2.0416194 2.3910572
## [208,] 0.7760904 1.5634969 0.7494969 0.7511937
## [209,] 2.2342440 4.8533254 1.9191719 2.1197792
## [210,] 1.5236116 3.0793804 1.1845187 1.4432187
## [211,] 2.2773615 4.5365250 1.8958257 2.0989432
## [212,] 3.5454869 7.5223232 3.7636957 4.4183222
## [213,] 2.7562394 4.7702179 3.0217535 2.6347039
## [214,] 1.3735486 2.7735531 1.3102935 1.3131781
## [215,] 4.0953556 9.1131269 3.3665457 4.7072203
## [216,] 3.8277157 7.0456135 3.3185580 4.7848382
## [217,] 1.1514626 2.1514237 1.4722582 1.4143003
## [218,] 1.1787620 2.6069467 1.1889876 1.1583109
## [219,] 1.7024884 3.3820649 1.8100894 1.7597684
## [220,] 1.9710872 4.0302314 2.1550237 2.5591027
## [221,] 2.2458071 4.3970831 2.1510478 2.4353259
## [222,] 2.0059158 4.0079085 2.0959498 2.0157237
## [223,] 2.0375596 4.2334419 1.9430534 2.2265719
## [224,] 1.3287732 2.4858355 1.5863467 1.7580503
## [225,] 5.1234648 9.3855567 5.7742356 6.0318599
## [226,] 1.4318833 2.6250286 1.2567433 1.7821632
## [227,] 1.0666295 2.2576435 0.9225259 0.8881553
## [228,] 3.3471947 6.1349819 3.0220101 2.8376161
## [229,] 2.2135136 4.0527839 1.8818837 2.8767735
## [230,] 1.9379279 3.0216216 2.4795401 4.0735135
## [231,] 2.3888772 4.6579866 2.2977769 2.0769059
## [232,] 1.9716805 3.2246776 2.3151358 2.8770211
## [233,] 1.0923760 2.3930252 1.4919160 2.3038028
## [234,] 5.4218375 11.9678781 4.4335299 6.2557304
## [235,] 1.6280206 2.4523753 2.1195743 3.5942355
## [236,] 1.7042224 3.3464818 2.3520191 2.7688231
## [237,] 0.5531993 1.2535208 0.5885674 0.5699880
## [238,] 2.3774607 4.4414865 2.2246483 2.6830857
## [239,] 2.0966392 4.4228362 1.8109470 1.7642007
## [240,] 1.0022125 1.6682513 1.3209404 1.8426466
## [241,] 1.9991763 3.9223921 1.8915651 2.1890776
## [242,] 3.8991073 8.4740926 4.3814504 5.8284797
## [243,] 2.6544714 5.0037877 3.5286955 3.4173593
## [244,] 1.0454837 1.8665756 1.1160844 1.1486061
## [245,] 3.2761475 6.4015569 3.3332802 3.5046783
## [246,] 1.4328072 2.7119202 1.1999425 1.3263253
## [247,] 3.3051851 7.8957211 3.2871203 3.3413147
## [248,] 3.6989912 6.7199784 3.4950511 3.2640728
## [249,] 1.2431365 2.0650912 1.2958791 1.2875548
## [250,] 2.1279117 4.7980532 1.8156600 2.7524151
## [251,] 1.1359590 1.9718272 1.1172831 1.0612553
## [252,] 1.0070151 2.0500910 0.8455926 0.8810514
## [253,] 2.3039549 4.2835543 2.4636977 2.9429260
## [254,] 0.4426458 0.9385151 0.3564292 0.4032716
## [255,] 1.8573378 3.1065595 1.9459607 2.2118295
## [256,] 2.8121403 4.6190492 3.0416272 3.5072071
## [257,] 3.1842899 6.4620906 2.8852620 2.8200109
## [258,] 1.3818004 2.2328747 1.4271479 1.5631903
## [259,] 1.0654907 2.0571381 1.1862238 1.1223487
## [260,] 0.5877676 1.2814954 0.6644516 0.6529047
## [261,] 1.3743475 1.9442641 1.9597070 3.2223579
## [262,] 2.5533848 4.8963783 2.1389439 2.5533643
## [263,] 0.3427253 0.6599216 0.3470824 0.3526053
## [264,] 2.3639295 4.7325300 2.1886641 2.0250764
## [265,] 0.7026909 1.2194473 0.8265812 1.1675271
## [266,] 2.6246953 5.4904393 2.3414538 2.0220735
## [267,] 2.5717283 5.3158766 2.9800994 2.4845983
## [268,] 2.0851706 4.4706064 2.1492826 1.9569465
## [269,] 1.6104680 2.9516023 1.4163498 1.9987043
## [270,] 1.7391978 3.8126931 1.7629392 1.8341632
## [271,] 2.3724004 4.1724561 2.5861564 3.2274244
## [272,] 2.2049438 4.5795117 2.4072726 1.8002861
## [273,] 0.9785817 1.7463176 0.8954782 1.1007411
## [274,] 1.6568943 3.0422607 1.8315826 2.3357141
## [275,] 1.5736036 3.2821064 1.4348801 1.6334048
## [276,] 0.1020453 0.2129584 0.1048847 0.1134031
## [277,] 3.0453578 5.8681208 2.8414837 3.0044817
## [278,] 1.7788002 3.8371927 2.1605704 3.2413217
## [279,] 1.2021875 2.3761667 1.4207323 1.5687560
## [280,] 2.1733422 4.7838996 2.3414339 2.8457091
## [281,] 1.8006604 3.2853727 1.8526549 2.0086384
## [282,] 2.5777716 5.2521621 2.1548615 1.9734987
## [283,] 1.5445357 2.8404568 1.5933904 1.7399544
## [284,] 2.6492747 4.5655682 3.0123773 4.1016849
## [285,] 0.6055747 0.9951957 0.7245288 1.0454268
## [286,] 1.9626782 3.5215629 1.9773908 1.9332532
## [287,] 3.5789467 6.0700769 4.5003946 6.2984880
## [288,] 2.4670586 5.4842081 3.2586440 5.4254566
## [289,] 4.6944135 9.3776903 5.2848669 5.8474784
## [290,] 3.3111493 6.5468262 2.9287782 2.9583977
## [291,] 2.6359566 4.5510399 3.0607869 3.2442325
## [292,] 2.1639457 4.8152857 2.0578551 2.3761269
## [293,] 1.1749776 2.6532597 1.4309832 1.9015010
## [294,] 1.3004214 2.5242174 1.3074268 1.3284428
## [295,] 1.6813067 3.2097775 1.3863006 1.8496381
## [296,] 3.6329658 5.5861094 4.6428002 7.6723035
## [297,] 1.2563178 2.4733213 1.2232533 1.3224469
## [298,] 1.8468835 3.6018733 1.8241495 1.7559476
## [299,] 2.8173103 5.9821113 3.0167133 2.4185042
## [300,] 1.3298925 2.2698296 1.5362925 1.7214950
## [301,] 1.6262458 2.7302794 1.7848221 2.2605508
## [302,] 4.3728122 8.8121084 4.2138881 4.2066021
## [303,] 1.9539168 4.0315831 1.9327087 1.9963328
## [304,] 4.4464866 8.7029516 5.3124791 4.9215404
## [305,] 2.2232574 3.9152105 2.2873159 2.4794915
## [306,] 2.5591903 5.5868495 2.7006785 3.1251430
## [307,] 1.9295090 4.2756117 1.5493808 2.2107371
## [308,] 2.1059265 4.7980303 2.3038098 2.4028426
## [309,] 2.2923357 5.1300184 2.0410100 2.7949871
## [310,] 1.7856368 3.3159619 1.8532824 2.0562193
## [311,] 0.9121039 2.0793370 0.9838739 1.1991838
## [312,] 2.2232003 3.4135404 2.8855164 4.8053510
## [313,] 2.8353736 4.6283972 3.0740279 3.7899907
## [314,] 1.8543632 4.0422766 1.4692133 1.7933920
## [315,] 2.4445932 5.6011443 1.9566813 3.0011909
## [316,] 1.0720573 2.4777295 1.4721602 2.6724688
## [317,] 1.8065511 4.2516054 1.4423127 2.3917351
## [318,] 1.7266419 3.3403220 1.7549049 1.8396938
## [319,] 1.8720836 3.2465315 1.7701541 2.0195152
## [320,] 2.7612909 5.7854212 3.3558980 4.6076035
## [321,] 4.4863715 8.8065960 5.0432064 4.9017391
## [322,] 2.8788677 6.4614671 2.6138733 3.3891263
## [323,] 1.6775148 3.4406308 2.0595249 2.3535602
## [324,] 2.4836396 4.8411933 2.7993806 2.7999067
## [325,] 2.8234126 5.6858582 2.8189987 2.8322406
## [326,] 1.6003668 2.9195687 1.6424127 1.7685503
## [327,] 0.6237266 1.2164813 0.5921595 0.6868608
## [328,] 1.9264356 3.5430987 1.7397071 1.5589093
## [329,] 1.5535062 2.6864439 1.6686879 2.0142329
## [330,] 0.6151655 1.3194515 0.5254922 0.6535487
## [331,] 1.4074941 2.7270676 1.7487232 1.6272958
## [332,] 0.6868676 1.2487450 0.6912572 0.7044257
## [333,] 1.1564374 1.7227524 1.5507540 2.6317433
## [334,] 3.2349197 6.2877173 2.7514140 2.6806474
## [335,] 1.1320722 1.9904802 1.1552653 1.0856862
## [336,] 2.4390703 4.5618267 2.0059966 3.2298553
## [337,] 2.7989600 5.7170523 2.6401215 2.7791138
## [338,] 2.4621835 5.0394350 2.0786483 2.1745131
## [339,] 1.2889954 2.6914923 1.2284709 1.0468974
## [340,] 1.5185371 2.8494719 1.7955181 1.6034936
## [341,] 2.0128111 4.0674304 1.9769637 1.9082905
## [342,] 3.1162255 6.0800249 2.7662448 2.0351989
## [343,] 1.2827917 2.8401572 1.5581437 2.3841999
## [344,] 1.3879958 2.8857009 1.4795445 1.7541909
## [345,] 4.5478640 9.1877211 3.6172419 4.0986986
## [346,] 2.2555539 4.4890312 1.9140967 2.3107457
## [347,] 1.4035464 3.3187773 1.5301131 1.9098135
## [348,] 3.3113070 7.5823022 2.6347021 3.7934231
## [349,] 3.5748108 7.4112782 3.0248260 2.9206695
## [350,] 2.5214494 5.3235593 2.3698493 2.8246497
## [351,] 2.7244173 5.1625209 2.1087688 3.3458460
## [352,] 4.2143944 8.6068651 3.3160064 3.7692038
## [353,] 1.9957394 4.1052878 1.8720440 1.7112168
## [354,] 0.9018702 1.7336200 0.9249564 0.8556980
## [355,] 0.6486773 1.4859367 0.9403580 1.7034941
## [356,] 1.3116781 2.3381437 1.4055830 1.5259767
## [357,] 1.3155634 2.8573141 1.2375618 1.4715664
## [358,] 0.4994221 1.0578980 0.4419423 0.4993325
## [359,] 3.2606373 6.0107854 2.8728662 3.0186203
## [360,] 1.1843927 2.3540183 1.2479157 1.2213104
## [361,] 3.5136261 6.6707123 2.9242846 3.5419042
## [362,] 3.0893551 5.9965694 2.9349071 3.3982511
## [363,] 3.7210322 6.3793167 3.5089053 3.6785157
## [364,] 2.2935836 5.0444281 2.9760999 4.8017686
## [365,] 1.4130076 2.6324096 1.6490687 1.7747583
## [366,] 0.5250795 1.0845322 0.5215646 0.5110199
## [367,] 0.7856890 1.3454356 0.9081899 1.2756926
## [368,] 0.4887277 0.9976707 0.5119699 0.5816966
## [369,] 2.0257123 4.2735980 1.7422294 1.7040556
## [370,] 2.1209502 4.2817057 1.8162836 2.0214368
## [371,] 1.2868996 2.8022405 1.2310511 1.0635055
## [372,] 1.4153793 2.8059756 1.3041735 1.1310100
## [373,] 1.4438789 3.2793207 1.9546169 3.4442623
## [374,] 1.6774837 3.5100254 1.9219100 1.4755217
## [375,] 2.3673966 5.6045996 2.2048735 2.5638282
## [376,] 2.0633133 4.2640831 1.9063279 1.7196718
## [377,] 1.7375319 3.2415700 1.3332729 2.3048465
## [378,] 2.3119852 4.7742247 2.3913438 2.6294195
## [379,] 0.3766412 0.8447030 0.4107394 0.5130339
## [380,] 1.6970938 3.3272591 1.4932113 1.8644150
## [381,] 2.0364470 4.0494995 2.0189302 1.9663799
## [382,] 2.1920237 4.2081502 2.1796093 2.1423661
## [383,] 2.5626254 5.2650196 2.3338744 2.2132030
## [384,] 0.8161236 1.6839236 0.6978488 0.6869324
## [385,] 0.8460489 1.7251798 0.7726250 0.7819062
## [386,] 1.2258521 2.5750455 1.1213540 1.3252090
## [387,] 1.1385302 2.0854815 1.2806130 1.1823079
## [388,] 1.2717867 2.7245989 1.3216819 1.4713675
## [389,] 1.1285624 2.2893985 1.2722703 1.3440438
## [390,] 1.1114886 2.0525144 1.3028854 1.3168280
## [391,] 2.0874846 4.2091435 2.0805466 2.0597325
## [392,] 1.5422246 2.6519045 2.1624708 2.5688327
## [393,] 3.0279183 6.3179854 3.3178278 3.5241164
## [394,] 1.8348045 2.9725579 1.8777912 2.0067515
## [395,] 0.9083854 1.6980907 1.2464461 1.2878064
## [396,] 2.6540875 4.8286382 2.7676095 2.5008698
## [397,] 3.1159384 6.0137777 3.2194342 2.9089469
## [398,] 1.5742591 2.6011006 1.8592378 2.0107113
## [399,] 4.7542518 9.9029413 4.3402113 3.7681579
## [400,] 3.2273625 6.5720603 2.8917262 2.8494284
## [401,] 1.8619130 3.5994345 2.0124799 2.1728917
## [402,] 2.1862704 4.7897369 2.3134280 2.6949010
## [403,] 2.0455541 4.0814580 2.3785619 2.6874441
## [404,] 0.9390214 1.9840584 1.3596945 2.0454068
## [405,] 2.6620800 5.1263140 2.6199662 2.7463075
## [406,] 1.0138195 1.5872638 1.1556524 1.5811508
## [407,] 0.8234429 1.7701087 0.9649671 1.3895398
## [408,] 2.2096766 3.9509729 2.1245366 2.3799565
## [409,] 2.2239509 5.0336288 1.7410557 2.7224782
## [410,] 1.2993611 2.6485412 1.6457296 2.1166453
## [411,] 3.3488010 7.2159606 2.9300765 2.5834744
## [412,] 1.7631660 3.2844339 1.7502230 1.7890520
## [413,] 1.9201577 3.7143883 1.6107296 2.2349758
## [414,] 0.8510589 2.0027248 0.8811686 0.9714978
## [415,] 1.5721085 3.1959187 2.0617633 2.4025967
## [416,] 2.1158874 3.4779959 2.4327543 2.6859779
## [417,] 1.3784279 2.8573711 1.7560727 2.3850059
## [418,] 1.5223646 2.7800475 1.4341538 1.6987862
## [419,] 0.3536191 0.7754592 0.3343908 0.3920757
## [420,] 3.4074526 7.4074803 3.2396133 2.7360954
## [421,] 3.0570682 5.9946077 2.3715097 3.0936219
## [422,] 1.0416666 2.1800145 1.5028956 2.2058278
## [423,] 0.9250623 1.6472347 1.0191374 1.1754397
## [424,] 2.9278443 6.4484705 3.9582610 6.3756294
## [425,] 0.8767075 1.7576116 1.0750198 1.1912772
## [426,] 0.6336637 1.5090806 0.5164478 0.8624119
## [427,] 1.7466791 3.3704389 1.5638679 1.4718965
## [428,] 2.2785520 4.2486133 2.1789079 2.4559742
## [429,] 1.3254361 2.5002276 1.3582852 1.4568324
## [430,] 5.3437955 12.3284313 4.2345059 7.2273171
## [431,] 2.3469378 4.5493137 2.1183942 2.7083427
## [432,] 3.5726848 7.9716109 3.3883492 2.8709053
## [433,] 5.1968467 11.7114959 4.2037351 6.4590003
## [434,] 0.7916710 1.7039196 0.6942119 0.8981970
## [435,] 1.1527572 1.8606107 1.2553123 1.5629776
## [436,] 0.9009433 1.9192710 0.7921435 0.9768048
## [437,] 1.8423142 3.3980272 1.6405136 1.9316368
## [438,] 1.5874993 3.2480729 1.9211180 2.3974761
## [439,] 1.0677478 1.8941107 1.2017795 1.6038748
## [440,] 2.8257221 6.3541626 2.6662368 3.1446928
## [441,] 1.5771411 2.7600822 1.8061998 2.3423682
## [442,] 0.8653561 1.6309638 0.9243851 1.1014722
## [443,] 3.7247704 7.7297106 3.4820108 3.9396761
## [444,] 0.4867962 0.7462417 0.6676576 1.0551731
## [445,] 3.3856534 7.7406251 3.1988063 3.7593474
## [446,] 0.5440531 1.0386135 0.6907678 0.6734300
## [447,] 1.4853366 2.9996234 1.3874800 1.4129611
## [448,] 2.1151717 4.4863817 2.7184473 3.8252527
## [449,] 0.8631578 2.0055325 0.9036313 1.0080001
## [450,] 1.5793088 2.8150514 1.6585574 1.8963031
## [451,] 0.5857299 0.9903382 0.6264502 0.7226633
## [452,] 2.2160212 4.9402526 1.9723090 2.7034457
## [453,] 2.0843133 4.3062755 1.6848363 1.6294818
## [454,] 2.5340160 4.8153044 2.8497891 3.1658348
## [455,] 3.7799091 8.2027165 3.2918577 4.4110517
## [456,] 2.2593070 4.4361607 1.9186623 1.6828055
## [457,] 1.6307558 3.3582478 1.4404530 1.3889155
## [458,] 0.9201378 1.7813243 0.7099257 0.9902755
## [459,] 0.7362718 1.6203059 0.6914687 0.5570593
## [460,] 1.5537183 3.0354142 1.5393672 1.5824207
## [461,] 4.9443239 9.5343835 4.4538658 4.1633996
## [462,] 1.5468303 3.4385938 1.2665730 1.8481747
## [463,] 1.6435786 3.7828033 1.3296307 2.2547981
## [464,] 0.8540987 1.7201758 0.9887388 1.1533249
## [465,] 1.8901390 4.0048115 2.2249873 3.1211695
## [466,] 2.6387698 5.7505469 2.8541873 3.5004398
## [467,] 3.5312130 6.4489319 2.9219426 4.2744249
## [468,] 0.8295650 1.7516611 0.9273123 1.1456108
## [469,] 1.6081459 2.3285182 2.1689841 3.5938119
## [470,] 1.1191945 2.1870614 1.3683849 1.4892565
## [471,] 4.0303857 8.1539018 4.0774074 4.2184727
## [472,] 2.3918011 5.0736961 2.1574282 1.6665662
## [473,] 1.3886818 3.1488212 1.5627209 2.0848380
## [474,] 2.5049872 5.5541151 2.3416815 2.2217351
## [475,] 2.6234997 5.3747439 2.2985030 2.2928678
## [476,] 2.7995714 5.8981716 2.8474563 2.7038015
## [477,] 0.8302197 1.5256460 0.7252180 0.7848566
## [478,] 2.2420973 4.5261315 1.7817698 2.1372553
## [479,] 1.5170575 2.7572030 1.4789069 1.3644551
## [480,] 1.1811894 2.2777109 0.9823995 1.3928592
## [481,] 3.7157250 8.5571299 3.0165097 4.9450554
## [482,] 1.8634726 3.5942088 2.4907514 2.7861892
## [483,] 1.1449724 2.6820133 1.1138794 1.0815339
## [484,] 0.3149336 0.6157796 0.2744531 0.3501524
## [485,] 2.2710650 3.9676738 2.4639899 3.0427643
## [486,] 3.2724808 6.1844314 2.8590907 4.0992610
## [487,] 3.1309414 6.6550998 3.2662708 3.6722593
## [488,] 4.0578598 7.3793102 3.1456908 5.8821978
## [489,] 2.2032937 4.4365623 2.3538203 2.1283563
## [490,] 0.3181738 0.6331994 0.3526642 0.3260446
## [491,] 0.8759801 1.6112340 0.8924973 0.8429459
## [492,] 1.5253827 3.0895658 1.7865318 2.1446817
## [493,] 1.0791932 2.2695568 1.3222500 1.4378610
## [494,] 1.7806119 3.0891719 1.6269299 2.0879760
## [495,] 0.9183674 1.9272671 0.9302991 0.8945040
## [496,] 3.3412328 7.3940427 2.8652867 4.1682832
## [497,] 2.8972803 5.0982921 3.0378505 2.8421310
## [498,] 2.6702070 5.0502774 2.8176142 3.2598359
## [499,] 1.6663300 3.5607654 1.4446433 1.7932203
## [500,] 1.1844345 2.4327507 1.2946968 1.5646633
## [501,] 3.9705844 7.8616592 3.8999677 4.1118178
## [502,] 0.9640942 2.1064690 1.0573901 1.3372781
## [503,] 1.1751468 2.2494347 1.1153866 1.2946673
## [504,] 4.5119525 9.5442458 3.5065168 3.8019333
## [505,] 2.4068596 4.3494398 2.5656113 2.8570608
## [506,] 1.6323890 3.7998916 1.2859040 2.2772029
## [507,] 1.4114389 2.8568018 1.5247638 1.3266288
## [508,] 5.8981465 12.7692385 5.4778098 4.4464393
## [509,] 1.1143594 2.3966324 1.3544185 2.0142620
## [510,] 1.9903979 4.4412279 2.6873492 4.5353804
## [511,] 0.5783272 0.9900777 0.6980459 0.9947688
## [512,] 3.5698480 6.8729127 3.2904515 2.4686466
## [513,] 1.1802895 2.5294173 1.2569180 1.4868035
## [514,] 1.3084965 2.4896264 1.3735118 1.5685579
## [515,] 1.9109573 4.1671846 2.3049699 2.4809665
## [516,] 3.0905677 5.6720314 2.6665383 3.6163767
## [517,] 0.8932856 1.6450210 0.9619268 0.7560033
## [518,] 2.7105450 5.4538702 2.9945969 3.3605994
## [519,] 2.7743699 4.9847732 2.1101401 4.0743210
## [520,] 0.9078595 1.6925258 0.8783228 0.7897127
## [521,] 1.5946867 3.6662830 1.4400170 1.8036294
## [522,] 0.3225677 0.6905576 0.4391126 0.5980831
## [523,] 3.0085863 5.2044637 3.3969222 4.5619300
## [524,] 1.6657866 3.6801100 1.7449503 1.9824413
## [525,] 2.8959381 4.7747691 3.8309563 5.4387061
## [526,] 1.7670178 3.6765121 1.6088434 1.8068603
## [527,] 0.8008658 1.6589246 0.7135810 0.6578835
## [528,] 6.0556753 13.1252270 6.1150421 5.9369418
## [529,] 1.8042739 3.5218170 2.0650242 2.0211011
## [530,] 1.8126076 3.3438987 1.6828697 2.0720835
## [531,] 2.0862188 3.7227307 2.4256792 3.2104862
## [532,] 2.2603466 5.1315987 2.2431217 2.2947964
## [533,] 1.8181648 3.2385577 2.2195954 2.8125944
## [534,] 0.6539955 1.3278406 0.7791848 0.9539983
## [535,] 2.9222105 6.5240016 3.2260122 4.1374172
## [536,] 4.7032006 8.5279846 4.6463187 4.4756732
## [537,] 2.3212424 4.5792740 1.9687496 2.4792694
## [538,] 1.2657825 2.8000596 1.2644081 1.2602850
## [539,] 2.9187531 6.0182917 3.0480710 3.4360249
## [540,] 1.8179320 3.6990230 1.4355425 1.6600347
## [541,] 1.3196440 2.4198475 1.7540521 1.8682453
## [542,] 1.3891208 2.6737473 1.4119528 1.4804488
## [543,] 1.9318495 3.2458900 2.3765215 3.3503298
## [544,] 0.8760239 1.6743714 1.0138068 0.9573984
## [545,] 1.3312708 2.8799262 1.5731047 1.7685874
## [546,] 1.8548012 3.1693571 1.9655156 2.2976587
## [547,] 3.2103046 7.2801957 2.6845739 3.6764377
## [548,] 1.3478349 2.3850919 1.4283942 1.6700721
## [549,] 0.8161478 1.5735655 0.9487146 0.9344560
## [550,] 1.6533052 3.0788478 1.9848808 2.2227117
## [551,] 1.9869956 4.1847901 2.1059110 1.7491649
## [552,] 3.2514208 5.1824345 3.6673377 4.9150883
## [553,] 1.7295846 3.6359454 1.8750977 1.9843759
## [554,] 1.0392104 2.0591470 0.9137721 0.7401490
## [555,] 0.9264999 1.9833548 0.8923283 0.9948430
## [556,] 3.1097067 6.1070998 2.9036584 3.1757568
## [557,] 4.3871239 7.6947341 4.4031002 4.4510291
## [558,] 0.7490194 1.3239251 1.0239476 1.0200181
## [559,] 2.5795135 5.7624758 2.4045824 2.0851581
## [560,] 2.3233498 4.6384417 1.7740034 2.2462611
## [561,] 1.0670345 2.0668741 1.0332003 0.9316979
## [562,] 3.0432513 6.6858315 3.8834771 5.9265163
## [563,] 0.2638877 0.4244853 0.3784269 0.5057162
## [564,] 3.3731182 7.8925169 3.8054328 5.1023766
## [565,] 5.1167938 10.0356075 4.5340316 5.6117443
## [566,] 5.4503594 10.7686913 4.9882231 5.7804280
## [567,] 0.3499051 0.7664771 0.3554486 0.3388180
## [568,] 1.7020448 3.5171623 1.7275147 1.6511051
## [569,] 0.8671603 2.0317380 0.8677566 0.8695455
## [570,] 0.3738865 0.7722987 0.3795330 0.3964723
## [571,] 3.2139286 5.5391306 3.4937757 4.3333170
## [572,] 2.0107456 4.2343312 1.6983277 1.9180100
## [573,] 2.3093514 4.7845547 3.1130129 4.3313044
## [574,] 1.1478046 2.0562459 1.2130109 1.0173921
## [575,] 3.5477469 5.6390574 4.4601801 7.1888377
## [576,] 2.0411294 4.6049834 1.6505050 2.3912912
## [577,] 0.4942774 0.8879876 0.5656318 0.7456950
## [578,] 2.2267776 3.9241960 2.1772134 2.0285210
## [579,] 2.3474365 3.7458803 2.7612632 3.5579410
## [580,] 2.2187659 3.7726893 2.6210918 3.5041498
## [581,] 1.5144938 2.4920101 1.6988331 2.2518510
## [582,] 1.9001224 3.9918551 2.4519239 3.0221521
## [583,] 0.6927968 1.5963865 0.9220025 1.4863106
## [584,] 1.6477242 3.6659284 1.5260964 1.8490902
## [585,] 2.5587924 5.0421052 2.2760211 2.4207232
## [586,] 1.4451837 3.3162408 1.2128622 1.7132381
## [587,] 3.8493575 7.3302841 4.6688824 4.7704346
## [588,] 3.5129218 7.5329201 3.3792697 2.9783133
## [589,] 1.6689534 3.1889537 1.7568171 1.4932259
## [590,] 1.1432744 2.4756820 1.0820419 0.8983443
## [591,] 1.8717141 3.9208692 1.8330310 1.7169820
## [592,] 2.0172135 4.2194170 1.6744169 1.7940952
## [593,] 2.3519778 4.9517174 2.4121419 2.2316495
## [594,] 2.3492892 5.1473367 2.7167823 3.8192614
## [595,] 1.4758466 2.8101481 1.4219222 1.5836955
## [596,] 4.1062195 7.8912610 4.4624166 4.9091646
## [597,] 2.2704141 4.3497462 2.5789860 2.4098529
## [598,] 1.4311685 2.8262164 1.2720714 1.5214700
## [599,] 5.4138231 12.1545298 4.2500023 6.0135372
## [600,] 1.9556166 3.8837013 1.5068108 2.0244462
## [601,] 1.0193693 2.3024309 0.9787709 1.1005660
## [602,] 3.7081045 7.2087463 4.4965137 4.2267612
## [603,] 4.6763988 8.5697367 4.3977348 4.1882076
## [604,] 2.4118585 4.8091030 2.2272312 2.4483937
## [605,] 4.2065464 8.4128058 3.8321491 3.4570344
## [606,] 1.4694123 2.6143004 1.3970645 1.6141080
## [607,] 2.0285716 4.4096888 2.6256065 3.8196133
## [608,] 0.5832373 1.1066650 0.6445955 0.5564296
## [609,] 2.5201836 4.6939489 2.4820027 2.5965454
## [610,] 3.0679174 4.9673442 3.2764465 3.9020340
## [611,] 2.7050729 6.2323734 2.6960811 2.7230565
## [612,] 1.7987549 3.7985413 1.5903023 1.4310131
## [613,] 1.8133658 3.5782546 1.8746262 1.9345582
## [614,] 1.9903456 3.3073689 2.2859227 2.6041966
## [615,] 1.2810162 2.6951666 1.0772674 1.2063540
## [616,] 1.1686299 2.4270969 1.2210354 1.3782520
## [617,] 4.7254071 8.8194064 3.6819569 6.2771519
## [618,] 3.3976505 7.3192326 2.6876171 3.2874128
## [619,] 1.5997065 2.9934275 1.6594126 1.4802943
## [620,] 1.7111513 3.1445638 1.5892695 1.9549149
## [621,] 2.6379270 5.3894311 2.1889086 2.3539841
## [622,] 2.3570349 5.1599594 3.1580609 4.3615028
## [623,] 1.8128639 3.7356005 1.3814548 1.5381823
## [624,] 0.9030712 1.5346851 1.1244120 1.5817142
## [625,] 3.5473941 6.7723076 3.3175775 3.2275239
## [626,] 2.7090148 6.0937257 2.8274369 3.0085957
## [627,] 1.3756609 1.9262987 1.9496542 3.4382184
## [628,] 1.5823912 2.9851245 1.9603913 2.0315358
## [629,] 0.7976840 1.7907669 0.6696091 1.0300315
## [630,] 1.9650143 4.3488418 1.7930869 2.3088690
## [631,] 5.2072521 11.7585146 4.3183953 6.7895644
## [632,] 3.5997059 6.7955785 3.2874512 4.2242153
## [633,] 2.5390044 4.4780612 2.5169025 2.4505971
## [634,] 2.7091129 5.1064777 2.3816100 3.3641185
## [635,] 2.3696293 4.8131457 1.9513840 2.1849118
## [636,] 2.3699433 4.6038361 2.2449646 1.8700285
## [637,] 1.5560693 3.3567900 1.3495099 1.4207714
## [638,] 2.1983852 4.4729147 1.8769122 2.0080246
## [639,] 2.2441376 4.6762140 2.5956868 2.8866649
## [640,] 1.8105070 3.5257289 1.7560130 1.9194951
## [641,] 1.1931368 2.5519818 1.0687452 0.7788663
## [642,] 1.9795421 4.5970250 1.6569024 2.3258914
## [643,] 1.3820859 3.2224326 1.1337307 1.7615171
## [644,] 2.3678850 4.2327345 2.5594984 3.1343384
## [645,] 1.6636229 3.2403134 1.8056484 1.7530532
## [646,] 1.4409943 3.1611892 1.5662519 1.9420246
## [647,] 5.6094583 11.0439187 5.7305286 5.4141043
## [648,] 2.2389558 4.6137914 2.6045379 3.3098192
## [649,] 1.1055302 2.3291177 1.2350568 1.6236368
## [650,] 0.9062267 2.0803748 0.6959956 1.1555682
## [651,] 4.7807099 8.9438626 3.7722457 5.9081720
## [652,] 0.9146803 1.7815124 0.9178173 0.9084063
## [653,] 3.5557164 6.6984924 3.9371043 4.5880676
## [654,] 1.9632814 3.8592593 2.5233021 2.2103364
## [655,] 0.9353066 1.6993340 1.1234763 1.3635049
## [656,] 2.9294820 5.5753264 2.7935008 2.3855571
## [657,] 0.7037711 1.5631172 0.8446096 1.2545940
## [658,] 0.9000070 1.8861269 1.0232217 1.3617187
## [659,] 2.0685051 3.4219784 2.6447612 3.8560848
## [660,] 2.2913027 4.9429928 3.0232831 3.8360399
## [661,] 3.0261817 6.1176310 2.7537743 2.7174165
## [662,] 1.3798878 2.7390772 1.3994881 1.3673423
## [663,] 2.6802319 5.7700547 2.6222981 2.4484967
## [664,] 0.6989502 1.3937060 0.7919038 0.7094363
## [665,] 1.0901873 2.1773337 1.1205870 1.0977899
## [666,] 1.7952549 3.5017162 1.6296524 1.6991268
## [667,] 3.1612028 6.1174676 4.1046144 4.5356812
## [668,] 3.6225559 8.4734665 3.9356866 4.8750787
## [669,] 2.3485093 4.4847344 1.9484656 2.8792198
## [670,] 0.8428480 1.9876408 0.8976417 1.0620229
## [671,] 1.0103727 1.9122744 0.9212580 1.1470587
## [672,] 4.3947762 8.1923152 4.8294509 5.8878692
## [673,] 3.0236165 5.6447636 3.4157529 4.0297901
## [674,] 3.0974985 6.6497433 2.8268691 3.0942003
## [675,] 0.8525628 1.7238801 0.8984669 0.9912574
## [676,] 0.3582200 0.7570884 0.3856490 0.4192291
## [677,] 1.0022697 2.3199557 0.9342114 1.0972188
## [678,] 2.8865243 6.3686300 2.4710720 3.5445733
## [679,] 1.5112295 2.9865406 1.9130641 2.0715304
## [680,] 2.0098340 4.0355596 2.1105637 1.9701047
## [681,] 0.4451274 0.7152772 0.5394102 0.8222586
## [682,] 0.9253584 1.7950215 0.9734986 1.0645964
## [683,] 1.2292689 2.6154431 1.5498800 2.2627545
## [684,] 2.3557122 4.4375978 2.0233026 3.0205315
## [685,] 0.8757890 1.6709937 0.8378984 0.7847662
## [686,] 2.4659940 5.3454865 1.9384645 2.3851322
## [687,] 2.6622951 5.7006888 2.6622809 2.6623233
## [688,] 3.1447468 6.7578094 3.1749576 3.2655902
## [689,] 4.1261452 7.9026504 3.6514494 3.6754725
## [690,] 1.5583054 3.5615021 1.3983810 1.5511422
## [691,] 3.5922492 7.0138286 3.5915968 3.5935538
## [692,] 2.3580885 4.1240645 2.4696546 2.1349562
## [693,] 0.7650671 1.1687110 0.9086525 1.2842242
## [694,] 1.9805786 3.6803026 1.7233270 2.1393423
## [695,] 1.4465555 2.9753912 1.2805238 1.3201927
## [696,] 4.1175915 7.9020788 3.4441594 4.9503523
## [697,] 1.5944743 3.0234792 1.4631864 1.8570500
## [698,] 1.6151077 2.6418098 1.7398689 1.8003574
## [699,] 5.6369096 12.8531865 4.4162325 6.8962401
## [700,] 0.6231110 1.1481130 0.5567087 0.6733454
## [701,] 2.4068341 5.0557137 2.3603106 2.4998810
## [702,] 1.4448857 2.4407237 1.5926348 1.8768659
## [703,] 1.4058671 2.3484972 1.6941405 2.5589604
## [704,] 1.3984948 3.2624109 1.1784194 1.8386456
## [705,] 0.4539639 0.9010448 0.5385201 0.4711713
## [706,] 1.2954015 2.7338006 1.7483534 2.5587996
## [707,] 3.6722132 7.6952676 3.0174430 3.0238797
## [708,] 1.2764054 2.0810747 1.8497777 2.4557458
## [709,] 1.8430322 3.8428254 1.9211177 2.1553744
## [710,] 1.3363468 2.5249683 1.2337264 1.5415875
## [711,] 1.8059456 3.3886568 1.4855103 2.3640316
## [712,] 1.7310444 3.0127515 1.9245626 2.5051174
## [713,] 2.9404780 6.4647294 2.5080241 3.0329774
## [714,] 1.2088096 2.0458401 1.7502526 2.0559228
## [715,] 1.2385854 2.7453868 1.0906268 1.3062675
## [716,] 2.4151343 4.6956270 2.3060103 2.6333823
## [717,] 2.7779656 6.1866752 2.1338315 2.9272501
## [718,] 2.3787640 5.3085364 2.8090619 4.0999555
## [719,] 2.7955250 5.5869685 2.9915812 2.8057288
## [720,] 1.0648245 2.4007599 1.1607962 1.4487114
## [721,] 0.5559805 1.0537752 0.5464950 0.5749517
## [722,] 2.0865493 3.7379373 2.0051453 2.2493572
## [723,] 5.9303530 11.7464088 4.7258335 6.1473474
## [724,] 1.5891396 3.0181031 1.9849582 1.9803361
## [725,] 1.4288999 3.2512458 1.2396558 1.6838738
## [726,] 1.5529105 3.3624136 1.4132165 1.8322984
## [727,] 4.1743509 6.3431840 5.8561432 9.1881455
## [728,] 1.4547127 2.7208940 1.2571137 1.4207650
## [729,] 3.1799495 7.2109339 2.8462849 3.4867342
## [730,] 2.5540008 4.6522884 2.7038764 3.0902141
## [731,] 2.8662756 6.4204825 2.2659465 3.0647711
## [732,] 2.0112694 3.4603722 2.4773026 3.4166857
## [733,] 0.9738200 1.9078837 0.9929262 1.0502447
## [734,] 1.6711429 3.8055692 2.3482803 4.1836260
## [735,] 0.8179490 1.7248889 0.8492581 0.9431855
## [736,] 3.5555245 6.8415699 3.4394864 3.7876008
## [737,] 3.6280438 8.0498628 3.5466111 3.6698436
## [738,] 2.2488029 4.4634507 2.3672677 2.4003450
## [739,] 3.4415544 7.3791922 3.2725164 2.7654025
## [740,] 1.0006833 1.7614431 0.9424744 0.8803278
## [741,] 1.4847116 3.0288268 1.4210926 1.5059828
## [742,] 2.7519114 5.2431986 2.9949922 3.4034720
## [743,] 0.8010333 1.3558564 1.0224906 1.4165589
## [744,] 3.1813836 6.5361303 3.2128998 3.3074483
## [745,] 2.9416369 5.7906321 2.3057005 3.1732409
## [746,] 1.4051141 2.5816112 1.5111999 1.8294572
## [747,] 1.7361476 3.4300775 1.8272623 1.8128328
## [748,] 3.0696874 6.6682286 2.8709935 3.4670752
## [749,] 1.5907604 2.8956831 1.8159334 2.3053549
## [750,] 1.6759125 2.7296176 2.0436274 3.0366153
## [751,] 2.7473824 6.0124497 2.7735307 2.8519754
## [752,] 1.8291913 3.7089006 1.7582420 1.8135879
## [753,] 0.7479310 1.5705334 0.7584684 0.7900806
## [754,] 1.7553962 3.3158799 2.1753618 2.1080463
## [755,] 0.9925844 2.1266200 1.1217768 1.5093540
## [756,] 2.2780806 4.5537208 2.3544276 2.4246738
## [757,] 1.9332557 2.7781498 2.5201205 4.1785815
## [758,] 2.1932179 4.8344058 2.1619054 2.2319433
## [759,] 2.6160022 5.9067547 2.2235360 2.8795162
## [760,] 2.6330563 5.3773496 2.3691754 2.3833869
## [761,] 1.6692647 3.0585031 1.6577761 1.6233102
## [762,] 2.3562424 4.3894178 2.2590053 2.5507166
## [763,] 0.6265559 1.1106411 0.7688283 0.9827327
## [764,] 1.2778046 2.4351082 1.3378676 1.1576787
## [765,] 2.0049337 2.9058312 2.8637357 4.7650240
## [766,] 0.8318734 1.6187524 0.9700803 0.9443594
## [767,] 1.4444127 2.6619633 2.0018585 1.9921487
## [768,] 0.9644042 1.2412335 1.3652242 2.5452725
## [769,] 0.8122947 1.3904799 1.0180347 1.1353186
## [770,] 2.7595464 5.3631971 2.1108748 2.9956392
## [771,] 4.7425254 8.5506710 5.5308891 5.7472860
## [772,] 1.5355528 3.4237298 1.3910620 1.7359559
## [773,] 0.9723646 1.9517702 0.9656864 0.9766107
## [774,] 1.7048486 3.6085038 2.0146231 2.8214144
## [775,] 1.4208067 2.6756895 1.2869556 1.6885090
## [776,] 1.5758999 2.6563863 1.5158220 1.6960557
## [777,] 3.3382228 6.9932135 2.7284551 2.5463029
## [778,] 2.4660301 5.1044574 2.1594657 2.0350369
## [779,] 1.6797339 3.1496539 1.6526883 1.5715514
## [780,] 1.8044728 3.7769035 1.7535850 1.9062484
## [781,] 0.7560088 1.2278491 0.9659670 1.4664302
## [782,] 0.9551517 1.8521072 1.0138157 0.9360038
## [783,] 1.8883668 4.4194035 1.6294030 2.2380746
## [784,] 0.4036238 0.8588708 0.3782768 0.4543177
## [785,] 2.6494853 4.9894938 2.1101939 3.4231773
## [786,] 0.9475744 1.9550629 0.9629202 1.0089575
## [787,] 1.4500088 2.8187984 1.6057897 1.5585225
## [788,] 2.9838305 6.2631805 2.5381260 2.8312201
## [789,] 1.7387006 3.4252728 1.6052293 1.3661545
## [790,] 1.0743688 2.1623951 1.2161201 1.0402251
## [791,] 1.3026471 2.9793136 1.2794270 1.3490874
## [792,] 1.2478565 2.4402796 1.0712997 1.0606348
## [793,] 1.6646986 3.5011995 1.4132978 1.5914024
## [794,] 0.9840738 1.7452096 1.1007698 1.1129526
## [795,] 2.2952359 4.8526370 2.8924713 4.1451194
## [796,] 1.3940045 2.6252914 1.5508190 1.7172885
## [797,] 1.5310913 3.2452357 1.5959056 1.7903485
## [798,] 1.8678629 3.9943082 1.8551886 1.8171655
## [799,] 1.6478963 3.0723718 1.4579303 2.0278284
## [800,] 1.9897577 4.3653155 1.9161705 2.1369322
## [801,] 0.7417146 1.4954433 0.8386448 0.9656106
## [802,] 2.4158468 4.4947365 2.1630221 2.0390408
## [803,] 1.1188462 2.2666178 0.9519703 0.8574080
## [804,] 2.3063274 5.2169869 1.9056216 2.6800060
## [805,] 0.4372146 1.0079896 0.3497747 0.5962360
## [806,] 0.4536732 0.9263782 0.4178714 0.4296493
## [807,] 1.0841272 2.1664658 1.1557553 1.2229115
## [808,] 1.8110882 3.2870031 2.1513054 2.6490216
## [809,] 1.5826506 3.1454100 1.3783064 1.2604976
## [810,] 2.8048258 6.0672308 3.5865243 5.5121709
## [811,] 2.9936873 6.6071885 2.4306955 3.3114905
## [812,] 3.0606483 6.2126696 2.4461234 2.8322157
## [813,] 0.7362176 1.4044253 0.8601485 0.8140547
## [814,] 2.3709074 4.6044626 1.8685146 2.4352649
## [815,] 0.4887315 1.0024989 0.4420083 0.4578748
## [816,] 1.9415137 3.5333559 1.8026334 1.6263576
## [817,] 2.8503621 4.3194705 3.4722888 5.3380691
## [818,] 2.0140085 4.2638894 1.7247562 1.7221241
## [819,] 1.3500979 2.9592050 1.2259162 1.2673202
## [820,] 0.5813601 1.3101100 0.5322847 0.5468143
## [821,] 1.1304961 2.0069365 1.3292069 1.5537698
## [822,] 0.6540718 1.0377067 0.8865224 1.1683749
## [823,] 0.3519987 0.7810730 0.3073564 0.4412833
## [824,] 0.7577032 1.6082853 0.8058243 0.9501878
## [825,] 5.0189163 11.1329166 5.1008036 5.3464654
## [826,] 3.2845483 5.9749688 3.2604900 3.3326647
## [827,] 1.7108792 4.0167047 1.3470109 2.3169670
## [828,] 2.5361400 4.4917990 2.3626562 2.8831075
## [829,] 3.6631928 8.3862767 3.2073383 4.5749017
## [830,] 1.5008436 3.0453855 1.4464768 1.5013553
## [831,] 2.2736459 4.1345044 1.9243194 2.7284398
## [832,] 3.6348520 7.8822395 2.9095398 3.5814147
## [833,] 0.4837227 0.9913983 0.4651749 0.5065095
## [834,] 4.2825003 9.1505597 3.9411104 4.9652800
## [835,] 2.3218012 4.1648291 2.8485475 3.5187342
## [836,] 2.1548705 4.5038676 2.7841636 3.2091101
## [837,] 0.5600419 1.1352662 0.5579430 0.5516461
## [838,] 0.6751938 1.4538787 0.9110268 1.2821646
## [839,] 0.8442675 1.6692597 0.8839846 0.8755134
## [840,] 1.6989497 3.5820882 1.8968137 2.4904057
## [841,] 3.3616443 6.8538998 3.0380110 3.0351162
## [842,] 3.4883114 6.0236658 4.5558909 5.2250903
## [843,] 1.9687474 3.3560155 2.0697702 2.3728386
## [844,] 0.9525371 2.1471509 1.0065049 1.0551939
## [845,] 0.9826819 2.0282605 1.1981107 1.3863701
## [846,] 1.5493067 3.2948320 1.3697985 1.5038180
## [847,] 1.1804118 2.0967079 1.2450252 1.4388654
## [848,] 3.4492654 7.2470430 3.0141601 3.4503351
## [849,] 1.0608784 2.0741025 1.1234740 1.1800139
## [850,] 0.8468311 1.7281334 0.7195795 0.7606534
## [851,] 2.8257020 5.0176963 2.8867231 2.7036598
## [852,] 1.8071433 4.0041651 1.5407293 2.2490113
## [853,] 1.8788786 3.9210984 2.1396309 2.8087364
## [854,] 0.3506246 0.7899072 0.4601077 0.7483130
## [855,] 1.2504249 2.5854546 1.1275085 1.2161042
## [856,] 2.4054430 4.9239227 2.5367380 2.9306232
## [857,] 3.6634688 6.8830960 4.3452009 3.9173289
## [858,] 1.7998737 3.1993974 1.4406167 2.5183877
## [859,] 1.1799759 2.0881321 1.0860187 1.3678902
## [860,] 1.9592918 4.2608862 1.7059117 1.6497788
## [861,] 1.1747422 2.5384492 1.0629490 1.3983285
## [862,] 2.4601802 4.4614770 2.4362144 2.5081118
## [863,] 3.4368562 6.2133495 3.8049724 4.9093210
## [864,] 0.7581645 1.4199608 0.7316778 0.6522177
## [865,] 0.3937487 0.6515705 0.4776158 0.7292169
## [866,] 1.0529473 1.7667305 1.4213975 1.8238487
## [867,] 2.5593570 5.1242084 2.6794360 2.8132512
## [868,] 1.3469819 2.5169283 1.8683106 1.9470506
## [869,] 1.1989660 2.0533511 1.2878272 1.5544109
## [870,] 0.8093108 1.4955139 0.9027994 1.1170802
## [871,] 1.3553560 2.4956657 1.2867910 1.4109669
## [872,] 3.3997350 5.7009769 3.2708222 3.6575606
## [873,] 2.1282221 4.2481607 1.9392098 1.9258202
## [874,] 1.7370438 4.0386757 1.7826341 1.9194050
## [875,] 0.7881376 1.6236026 0.6655348 0.6698193
## [876,] 0.7739259 1.4025476 1.0009720 1.1362921
## [877,] 0.3077894 0.6480820 0.4323214 0.6381112
## [878,] 1.1081456 2.5319792 1.1571280 1.2449328
## [879,] 4.6597515 10.6864179 3.5835106 5.9245571
## [880,] 3.4994814 8.0505893 3.0517901 4.3948638
## [881,] 2.0234900 4.5199292 1.7477895 2.5748911
## [882,] 3.6547005 7.4272116 3.1709564 3.3601738
## [883,] 0.7704341 1.6223512 0.7920293 0.8568150
## [884,] 2.8548291 5.7441800 2.9129142 2.7685247
## [885,] 2.7392900 4.5480702 3.1903883 4.5436829
## [886,] 2.9447177 4.2137734 3.7716426 6.0512229
## [887,] 1.0128638 2.2564567 1.1125631 1.4116607
## [888,] 1.8468849 3.2529773 2.1730030 2.9488658
## [889,] 3.2966667 6.4207765 3.0510488 2.8583388
## [890,] 2.9054433 4.9417660 2.6966989 3.0602917
## [891,] 1.1023896 2.3732386 0.8752884 1.0693354
## [892,] 1.6088309 3.5573482 1.5775272 1.6354181
## [893,] 1.3373382 2.7357988 1.1296066 1.0746808
## [894,] 2.7232307 5.5297525 2.9460029 3.3770026
## [895,] 1.3974086 2.6500973 1.2157758 1.7592083
## [896,] 1.3105544 2.4904482 1.4384947 1.6372063
## [897,] 0.9262159 1.6529077 0.9903862 1.0018069
## [898,] 2.0348530 4.2850624 2.4381486 2.8943446
## [899,] 4.5930611 9.1458222 3.8937487 3.5984243
## [900,] 0.4987117 0.7959174 0.6101927 0.8549725
## [901,] 0.9740924 2.0424803 0.9307469 0.8401296
## [902,] 1.2763227 2.6521500 1.4824560 1.9373507
## [903,] 2.0858491 3.4077335 2.2024434 2.3265423
## [904,] 1.0922717 2.0385337 1.3344671 1.2116383
## [905,] 1.8382176 3.6885158 1.9989828 2.1899492
## [906,] 1.2890075 2.7931424 1.3433890 1.5065333
## [907,] 1.0752620 2.0182661 1.0539997 1.1142139
## [908,] 0.9810990 1.7398407 1.0987382 1.4516557
## [909,] 2.9961644 4.9667975 3.8586435 5.5599927
## [910,] 3.3047304 7.1822569 3.1017609 2.7604354
## [911,] 1.9991588 3.6258127 1.8611848 1.7435253
## [912,] 1.9848667 3.5097437 2.2618563 2.4281073
## [913,] 3.2985888 6.6390560 3.2169517 3.1938930
## [914,] 2.5578317 4.4637392 2.6307320 2.8089727
## [915,] 0.9644532 2.1670194 0.8972897 1.0987804
## [916,] 0.7141368 1.4853712 0.6736356 0.5713931
## [917,] 3.6252403 6.7011988 2.9955063 4.8847083
## [918,] 1.7046497 2.9369011 1.9763234 2.7913448
## [919,] 0.6461710 1.0038839 0.8054629 1.2833383
## [920,] 0.9612060 1.9275915 0.9003154 0.9482574
## [921,] 0.3648473 0.7607258 0.4221722 0.5570751
## [922,] 1.8537456 3.6961332 2.2189658 2.5557910
## [923,] 1.5342492 2.9819481 1.5410405 1.5206666
## [924,] 3.0328758 4.8073701 3.6772039 5.6101880
## [925,] 3.6665593 6.4602762 3.2096254 4.1049464
## [926,] 1.3676128 2.2922266 1.5067381 1.6778606
## [927,] 2.8666478 5.7107846 3.5977360 4.1461955
## [928,] 1.0739950 2.0668831 0.9219197 1.2767624
## [929,] 4.5909351 10.3484143 4.5034313 4.6113907
## [930,] 1.2102660 2.2606539 1.2306901 1.1694178
## [931,] 2.2282756 4.3280779 2.0041864 2.5494585
## [932,] 2.8887218 5.8125222 2.7445427 2.6880599
## [933,] 1.3268583 1.7728971 1.8661569 3.4840524
## [934,] 2.3920286 5.1697430 2.5726057 3.1143369
## [935,] 1.3832062 2.9773151 1.2901040 1.5694105
## [936,] 1.9586393 4.3338292 1.5445997 1.9142646
## [937,] 3.0524233 6.7960892 2.7439287 3.6694125
## [938,] 2.0547712 4.1288693 2.5371949 3.0679361
## [939,] 0.6443076 1.3525224 0.6457519 0.6500848
## [940,] 3.7563294 7.0551795 3.3897745 4.4894392
## [941,] 1.3494729 2.5277157 1.4346576 1.6902118
## [942,] 1.4993020 2.6973462 1.2824822 1.8281298
## [943,] 0.3622994 0.7176281 0.3727424 0.3657585
## [944,] 1.5650340 1.9439842 2.2995290 4.5030140
## [945,] 1.8371196 3.5736370 1.7281951 1.7647377
## [946,] 1.6364151 3.0895316 1.7593617 2.0946619
## [947,] 2.3119657 4.4904154 2.2610275 2.4138422
## [948,] 1.5843382 3.3822304 1.4540654 1.0632472
## [949,] 0.8532140 1.5849595 1.0107372 1.1568850
## [950,] 0.5884748 1.0395352 0.5732103 0.6190038
## [951,] 0.9242687 2.0666359 0.7280898 1.0771573
## [952,] 1.9260943 3.5068814 1.6405784 2.0313572
## [953,] 1.0492157 1.9540757 1.0041419 0.8689204
## [954,] 1.8902323 3.1632314 2.3579464 3.2840301
## [955,] 2.9465113 4.5577334 3.3380841 4.5128023
## [956,] 3.6971624 8.4268580 3.0926787 4.5350785
## [957,] 0.8454645 1.9892942 0.8428008 0.8507919
## [958,] 2.8087297 6.1702253 2.9831732 2.7133844
## [959,] 1.4942319 3.1175024 1.5274828 1.4277302
## [960,] 3.0248475 5.8461843 3.2805871 3.5336246
## [961,] 0.5784881 1.3189105 0.6591278 0.8784514
## [962,] 2.5134929 5.7467552 2.3591161 2.8222466
## [963,] 1.0402812 2.2982278 0.9115936 1.2976566
## [964,] 2.5234370 4.7074786 2.8777880 2.3836509
## [965,] 1.0139333 1.6669177 1.3310797 1.9163052
## [966,] 1.7922228 3.2254853 1.6608340 1.7359260
## [967,] 4.3539031 8.6233275 3.9937188 3.4223378
## [968,] 1.5172450 3.2927974 1.3015325 1.7315888
## [969,] 1.4071734 2.2529503 1.7629279 2.8106645
## [970,] 1.6898442 3.3895553 1.3189758 1.6651773
## [971,] 2.6785035 5.3666054 2.8208897 2.9872718
## [972,] 1.7483176 3.7004121 1.7041644 1.5717045
## [973,] 1.8926139 4.4055844 1.8377892 2.0022633
## [974,] 1.3732813 2.6094192 1.2697240 1.2630236
## [975,] 1.4827564 3.2004520 1.5544498 1.7695300
## [976,] 1.4074224 2.4756080 1.7689884 1.6152998
## [977,] 5.7834314 12.1125069 4.9188602 5.4183991
## [978,] 0.8199017 1.4154883 0.9226607 1.2309377
## [979,] 0.8999243 1.8693285 0.9639301 0.7719127
## [980,] 3.3628141 6.3114785 3.7408982 4.3981883
## [981,] 0.9764564 1.9149714 0.9872207 1.0195136
## [982,] 1.9046817 3.7838470 2.3353626 2.7022523
## [983,] 1.4211344 2.8791595 1.8106179 2.2923284
## [984,] 2.2325574 4.0245395 2.1439535 1.8781417
## [985,] 1.1972978 2.3325661 1.2224408 1.2978700
## [986,] 1.5715896 2.8518614 1.6943538 2.0626464
## [987,] 2.1465470 4.4101097 1.9405009 1.8597003
## [988,] 0.9458160 1.8606026 0.8897091 1.0233895
## [989,] 3.4025307 7.1685533 2.8905175 3.2872343
## [990,] 0.5830169 1.2742918 0.5507494 0.4539468
## [991,] 1.8860541 3.5353024 1.8033999 1.7260341
## [992,] 2.1088647 4.1312991 1.9766026 1.6282650
## [993,] 1.7034966 2.9872770 1.7405004 1.8515116
## [994,] 2.4421993 4.1977976 2.7877378 3.4462672
## [995,] 4.2146361 9.3630739 4.1774992 4.0660882
## [996,] 2.5631327 4.8618388 2.9113633 3.2241993
## [997,] 0.8877602 1.8578264 1.1179292 1.5170819
## [998,] 1.0462054 2.1320443 1.0956247 1.2438829
## [999,] 2.8955092 6.3151204 3.5267204 5.4203541
## [1000,] 2.9179551 6.1031171 2.6072665 2.6218364cat("Media \n")## Mediaprint(prom<-apply(resultado,2, mean)) ## Estimadores1 Estimadores2 Estimadores3 Estimadores4
## 2.008738 4.003790 2.017780 2.363169# valores de las medias
cat("Desviaciones \n")## Desviacionesprint(desv<-apply(resultado,2, sd)) ## Estimadores1 Estimadores2 Estimadores3 Estimadores4
## 1.073384 2.236487 1.029477 1.324391# valores de la desviaciones estándar
boxplot(resultado, col = c("purple","pink","green","yellow"), ylim = c(0,10),main = "Estimadores " , ylab="Estimacion")
abline(h=theta,col='red') # línea indicando 1/2
grid()
Conclusiones:
Después de realizar 1000 repeticiones en la primera ejecución, se pueden extraer las siguientes conclusiones sobre los resultados obtenidos de los 4 estimadores:
Respecto a la insesgadez, se observa que los estimadores 1 y 3 son los más cercanos al valor de theta (2). Sus resultados fueron 2.00 y 2.01 respectivamente, lo que sugiere una mayor aproximación al valor real.
En cuanto a la eficiencia, se encontró que los estimadores 1 y 3 presentan una menor varianza, con valores de 1.07 y 1.0 respectivamente. Esto indica una mayor consistencia en sus resultados y una menor dispersión con respecto al valor estimado.
El estimador 2 se identificó como el menos confiable debido a que tanto su insesgadez como su varianza muestran valores negativos. Su media fue de 4, alejándose considerablemente de theta, y una desviación de 2.23, la más alta entre los estimadores, lo que sugiere una mayor variabilidad en sus resultados.
En relación con el estimador 4, aunque sus valores no son alarmantes, no alcanzan la óptima precisión de los estimadores 1 y 3. Su media fue de 2.36 y la desviación de 1.32, lo que indica una menor consistencia y precisión en sus estimaciones.
Conclusiones Generales:
Los estimadores 1 y 3 destacan por su mayor aproximación al valor de theta, menor varianza y mayor eficiencia en sus estimaciones.
Mientras que el estimador 2 muestra resultados menos confiables.
El estimador 4 se encuentra en un punto intermedio entre los anteriores.
Es importante destacar que todos los estimadores se calcularon con la misma cantidad de valores, lo que asegura la propiedad de consistencia en el método utilizado para obtener los resultados.
A medida que se realizan mas repeticiones las diferencias se van atenuando.